Abstract: We study two-dimensional $\mathcal{N}{=}(0,2)$ supersymmetric gauged linear
sigma models (GLSMs) using supersymmetric localization. We consider
$\mathcal{N}{=}(0,2)$ theories with an $R$-symmetry, which can always be
defined on curved space by a pseudo-topological twist while preserving one of
the two supercharges of flat space. For GLSMs which are deformations of
$\mathcal{N}{=}(2,2)$ GLSMs and retain a Coulomb branch, we consider the
$A/2$-twist and compute the genus-zero correlation functions of certain
pseudo-chiral operators, which generalize the simplest twisted chiral ring
operators away from the $\mathcal{N}{=}(2,2)$ locus. These correlation
functions can be written in terms of a certain residue operation on the Coulomb
branch, generalizing the Jeffrey-Kirwan residue prescription relevant for the
$\mathcal{N}{=}(2,2)$ locus. For abelian GLSMs, we reproduce existing results
with new formulas that render the quantum sheaf cohomology relations and other
properties manifest. For non-abelian GLSMs, our methods lead to new results. As
an example, we briefly discuss the quantum sheaf cohomology of the Grassmannian
manifold.